1.1.1 Goals of this lecture

• Mixed model for categorical outcomes
  • Marginal model: \(\textbf{R}\) matrix, GEE
  • Conditional model: \(\textbf{G}\) matrix, GLMM

2.1.1 Repeated measures = non-independence

• Repeated measures from the same person are not independent
  • An observation from a person provides information about other observations from that person
  • Observations from the same person are more like one another than observations from different people
  • Observations from the same person are correlated

2.1.2 Linear mixed models

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Very general model
• Allows for repeated measures via
  • Random effects: \(\mathbf{Z}\mathbf{\gamma}\)
  • Correlated residuals: \(\boldsymbol{\epsilon}\)

2.1.3 Linear mixed model: Marginal approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Fixed effects: \(\mathbf{X}\boldsymbol{\beta}\)
  • \(\mathbf{X}\) is a matrix of the predictors
  • \(\boldsymbol{\beta}\) are regression coefficients

2.1.4 Linear mixed model: Marginal approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Random effects: \(\mathbf{Z}\mathbf{\gamma}\)
  • No random effects
    • No predictors in \(\mathbf{Z}\)
    • This term drops out

2.1.5 Linear mixed model: Marginal approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Residuals: \(\boldsymbol{\epsilon}\)
  • Residuals are correlated / covary across timepoints
    • \(t \times t\) matrix

2.1.6 Linear mixed model: Conditional approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Fixed effects: \(\mathbf{X}\boldsymbol{\beta}\)
  • \(\mathbf{X}\) is a matrix the predictors
  • \(\boldsymbol{\beta}\) are regression coefficients

2.1.7 Linear mixed model: Conditional approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Random effects: \(\mathbf{Z}\mathbf{\gamma}\)
  • \(\mathbf{Z}\) is a matrix of random effects predictors (dummy codes)
    • Which observations go with which subject
  • \(\mathbf{\gamma}\) is a variance-covariance matrix of random effects
    • Intercept variance, slope variance, intercept-slope covariance

2.1.8 Linear mixed model: Conditional approach

\[\mathbf{Y}_{ij} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{\gamma} + \boldsymbol{\epsilon}\]

• Residuals: \(\boldsymbol{\epsilon}\)
  • Residuals aren’t correlated
  • Single residual variance value
    • Variance of all residuals across all people and timepoints

2.1.9 Marginal vs conditional

• Deal with non-independence in different ways: Different interpretation
  • Marginal: Effect based on all observations, then adjust standard errors
  • Conditional: Effect for each subject, then average across subjects
• For linear models, marginal and conditional don’t differ numerically
  • Average of linear function = linear function of average

2.1.10 Marginal vs conditional

• For non-linear models, marginal and conditional differ numerically
  • Average of a non-linear function \(\ne\) non-linear function of average
  • Remember from GLiM: Non-linear function of predicted value \(\ne\) predicted value of non-linear function
• Again, that’s ok
  • Marginal and conditional models are actually answering different questions

2.1.11 Comparison

3.1.1 Schizophrenia over time

• Schizophrenia treatment effects over the course of 7 weeks (\(N\) = 437), measured by the Inpatient Multidimensional Psychiatric Scale (IMPS)
  • id: ID variable
  • imps79: Continuous measure of schizophrenia (1 to 7)
  • imps79b: Binary measure of schizophrenia (3.5+)
  • imps79o: Ordinal measure of schizophrenia (Cuts: 2.5+, 4.5+, 5.5+)
  • tx: Placebo (0) or treatment (1)
  • week: Week of study (0, 1, 3, 6)

3.1.2 Data

3.1.3 Means and \(N\) by week

3.1.4 Plot: Continuous measure by week

3.1.5 Plot: Binary measure by week

3.1.6 Plot: \(\color{red}{Marginal}\) effect of time

3.1.7 Plot: \(\color{blue}{Conditional}\) effect of time

3.1.8 Plot: \(\color{red}{Marginal}\) and \(\color{blue}{conditional}\) lines

3.1.9 Plot: \(\color{red}{Marginal}\) and \(\color{blue}{conditional}\) lines

3.1.10 Overall model

• Research question: Now
  1. How does schizophrenia diagnosis change over these 7 weeks?
• Research question: Class
  1. How does schizophrenia diagnosis change over these 7 weeks?
  2. How do the treatment groups differ (at baseline)?
  3. Does change in diagnosis differ depending on treatment condition?

4.1.1 Marginal model: Generalized estimating equations (GEE)

\[\eta = \mathbf{X}\boldsymbol{\beta}\]

• \(\eta\): Transformation of predicted value (from GLiM)
  • Depends on the specific model (i.e., logistic, Poisson)
• Variance
  • \(\epsilon\): Matrix of correlated residuals
    • \(t \times t\) matrix: \(t\) is number of repeated measures

4.1.2 Marginal model: Generalized estimating equations (GEE)

• Fixed effects (regression coefficients)
  • Population-averaged effects
    • Averaging across all observations (ignore people)
  • For “people”, not for “a person”
    • Public health application

4.2.1 Example

• Time predicts schizophrenia diagnosis
  • week as a predictor of imps79b
  • week: 0, 1, 3, 6
  • imps79b: 0 (less than 3.5 on imps79), 1 (3.5+ on imps79)
• In this example, I’m using unstructured \(\textbf{R}\) matrix
  • We’ll look at others in class
  • We’ll also look at treatment effects (tx)

4.2.2 Marginal model

Call:
geeglm(formula = imps79b ~ 1 + week, family = binomial("logit"), 
    data = schizx1, id = schizx1$id, corstr = "unstructured")

 Coefficients:
            Estimate  Std.err  Wald            Pr(>|W|)    
(Intercept)  2.59459  0.11876 477.3 <0.0000000000000002 ***
week        -0.45017  0.02767 264.7 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation structure = unstructured 
Estimated Scale Parameters:

            Estimate Std.err
(Intercept)   0.9646  0.1007
  Link = identity 

Estimated Correlation Parameters:
          Estimate Std.err
alpha.1:2  0.05390 0.05343
alpha.1:3 -0.02855 0.03265
alpha.1:4 -0.01890 0.03342
alpha.2:3  0.56341 0.10114
alpha.2:4  0.15242 0.06617
alpha.3:4  0.51550 0.07964
Number of clusters:   437  Maximum cluster size: 4 

4.2.3 Plot: Marginal model

4.3.1 Population-averaged effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week)\]

• \(b_1\): Time effect
  • \(e^{b_1}\) = \(\frac{odds~of~event~at~week~t+1}{odds~of~event~at~week~t}\)
  • Ignoring the repeated measures
    • But standard errors are adjusted for non-independence

4.3.2 Population-averaged effects: Pros

• Robust to mis-specification of the \(\textbf{R}\) matrix
  • You use compound symmetry but that’s not very close to reality
• Can account for unobserved or unknown dependence
  • Things besides repeated measures
• Easier to estimate than conditional models
  • Marginal values are readily available

4.3.3 Population-averaged effects: Cons

• Ignores that individuals make up these effects
  • Just wants to deal with correlated observations
• Ignores that individuals may have different patterns over time
  • Are some individuals helped a lot by the treatment? Who knows.
• Documentation for newer R package says it requires complete data
  • Worked fine here so ???
  • You can delete all NA rows, but an additional step

4.3.4 Why is it called “marginal”?

• Estimated using only marginal proportions
  • Not joint proportions: Therefore no conditional values either

5.1.1 Conditional model: Generalized linear mixed model

\[\eta = \mathbf{X}\boldsymbol{\beta}\]

• \(\eta\): Transformation of predicted value (from GLiM)
  • Depends on the specific model (i.e., logistic, Poisson)
• Variance
  • \(\gamma\): Matrix of random effects
    • Intercept variance, slope variance, intercept-slope covariance
  • \(\epsilon\): Residual variance
    • Single number (depends on model: e.g., fixed at \(\pi^2/3\) in logistic)

5.1.2 Conditional model as multi-level model

• Two parts (“levels” for multi-level models) of the model
  • Within-person / within-cluster
  • Between-person / between-cluster
• Equations at each level
  • Combine into the full model

5.1.3 Conditional model as multi-level model

• Level 1: Within-person equation
  • \(\eta = \pi_{0i} + \pi_{1i}(week) + e_{ij}\)
• Level 2: Between-person equation
  • \(\pi_{0i} = \beta_{00} + \color{blue}{r_{0i}}\)
  • \(\pi_{1i} = \beta_{10} + \color{blue}{r_{1i}}\)
• Combined equation
  • \(\eta = \beta_{00} + \beta_{10}(week) + \color{blue}{r_{0i}} + \color{blue}{r_{1i}}(week) + e_{ij}\)

5.2.1 Example

• Time predicts schizophrenia diagnosis
  • week as a predictor of imps79b
  • week: 0, 1, 3, 6
  • imps79b: 0 (less than 3.5 on imps79), 1 (3.5+ on imps79)
• In this example, we have random intercepts and random slopes
  • We’ll also look at treatment effects (tx) in class
  • For that model, we’ll only be able to use random intercepts

5.2.2 Conditional model

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: imps79b ~ 1 + week + (1 + week | id)
   Data: schizx1

     AIC      BIC   logLik deviance df.resid 
  1291.6   1318.4   -640.8   1281.6     1564 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8446  0.0898  0.1139  0.2646  1.0822 

Random effects:
 Groups Name        Variance Std.Dev. Corr 
 id     (Intercept) 4.413    2.101         
        week        0.711    0.843    -0.13
Number of obs: 1569, groups:  id, 437

Fixed effects:
            Estimate Std. Error z value            Pr(>|z|)    
(Intercept)    4.386      0.539    8.14 0.00000000000000041 ***
week          -0.793      0.118   -6.71 0.00000000001954283 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
week -0.817

5.2.3 Plot: Conditional model

5.3.1 Person-specific effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week)\]

• \(b_1\): Time effect
  • \(e^{b_1}\) = \(\frac{odds~of~event~at~week~t+1}{odds~of~event~at~week~t}\)
  • Estimated separately for each person
    • Average individual effects together to get the average effect

5.3.2 Person-specific effects: Pros

• Individual trajectories are estimated
  • Not just correlations between repeated measures
• More flexibility in individual variability
  • Random intercepts and random slopes with respect to time
• Conceptually, fits better with how psychologists think
  • Individuals, trajectories, etc.

5.3.3 Person-specific effects: Cons

• Often harder to estimate
  • Much more complex model than marginal model
• Deciding on random effects can be difficult
  • Both choosing and estimation
• Accounts for specific sources of non-independence
  • Cannot account for e.g., multiple members of same family

6.1.1 Example: Conditional vs marginal effects

• Marginal effect
  • For people, one week of time has a certain effect
  • Population level: Interest is the group of people
• Conditional effect (GLMM):
  • For a person, one week of time has a different effect
  • Individual level: Interest is individuals

6.1.2 Plot: Comparison of \(\color{red}{marginal}\) and \(\color{blue}{conditional}\)

7.1.1 Summary of this week

• Extended mixed models to categorical outcomes
  • Marginal: \(\textbf{R}\) matrix, population averaged, GEE, cluster robust
  • Conditional: \(\textbf{G}\) matrix, generalized linear mixed models (GLMM)
• Different interpretations, different numbers
  • Population-averaged: Adjusts for non-independence, doesn’t care about repeated measures
  • Conditional: Analyzes each person separately and averages

7.1.2 In class

• Look at models including week, tx, and their product
  • Think about how that more complex model works
  • See some errors you might get

7.1.3 Next week

• All Some of the additional crucial details
  • Estimation issues
    • Maximum likelihood?
    • Tips and tricks to get models to run?
  • Model comparisons
  • Multi-level issues: Adding and centering predictors, contextual effects
  • More on predicted values